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< 


NOVEMBER 1961 







PRINCIPLES OF ATMOSPHERIC REENTRY 


By 

Theodore A. George 


* * * 


Advanced Research Projects Agency 
Office of the Director of Defense Research and Engineering 

Washington 25, Do Co 


November 1961 






I 

D 



) 

i 



i 


CONTENTS 


Page 

Abstract- v 


1. Introduction- 1 

2. Fundamental Principles- 4 

2. 1 Celestial Mechanics- 4 

2. 2 Analysis of a Space Vehicle’s Energy- 5 

2. 3 The Earth’s Atmosphere- 5 

2. 4 Dynamics of Reentry- 8 

2. 4. 1 General Equations of Motion- 8 

2. 4. 2 Solution of the Equations of Motion- 9 

2. 4. 3 Effect of Reentry Trajectory- 15 


3. Aerodynamic Heating- 23 

4. Deceleration- 30 

5. Cooling Techniques- 39 

5. 1 Introduction- 39 

5. 2 Reflectance- 39 

5. 3 Transpiration Cooling- 40^ 

5.4 Ablation- 44 

5. 5 The Heat Sink- 45 

5. 6 Possible New Methods of Cooling- 46 

5. 6. 1 Thermal Insulation- 46 

5. 6. 2 Internal Cooling- 46 

5.6.3 Chemical Surface Reaction- 47 

5. 6. 4 Point Mass Addition- 47 

5. 7 Testing and Simulation- 48 

6. Space-Vehicle Configurations- 50 

7. Conclusions- 53 


Bibliography 


57 


Appendix I. Notation 


83 


111 


































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# 


ABSTRACT 

This paper summarizes the entire field of a space 
vehicle's reentry into the earth's atmosphere. It is 
assumed that the vehicle, either lifting or nonlifting, is 
approaching the earth from outer space and must pass 
through all densities of the atmosphere from /^ = O 
as it enters at high altitude until it lands at approximately 
sea level or P - (that is, ^ma/ )• 

During the vehicle's descent through the earth's 
atmosphere, deceleration must not exceed a maximum 
value, the total heat taken in by the vehicle must not 
be excessive, and the vehicle's skin or internal temper¬ 
ature must be limited. Methods of attaining these 
objectives are explained. 


V 




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1. INTRODUCTION 


The purposes of this paper are (1) to review, consolidate and, 
where appropriate, summarize available knowledge affecting the be¬ 
havior of a body entering the earth's atmosphere at a high velocity and 
(2) to indicate how the difficulties encountered during such a flight can 
be overcome. It must be emphasized, therefore, that this is not merely 
a summary of various technical papers on reentry but is rather an 
effort to present the entire subject in a single cohesive treatise that 
bridges existing gaps between the related disciplines. 

From a practical standpoint, this field of research has been con¬ 
cerned primarily with man-made objects (vehicles) in their plunge 
back into the earth's atmosphere; hence, the word "reentry" is usually 
associated with the subject. In January 1959, when this paper was 
begun, there was no single text to which a reader could refer for a 
general understanding of the complex field of atmospheric reentry. 

There were a large number of papers touching on certain specialized 
aspects of reentry, and most of these presupposed that the reader had 
a considerable background and understanding of the technical field 
covered. This situation made it very hard for the average engineer to 
obtain a broad view of reentry technology in a short time. Also, 
specialists working in the other areas of space technology--propulsion, 
communications, celestial mechanics --frequently need a dependable 
reference on reentry when their work overlaps that field. This paper 
attempts to satisfy these requirements; also, an exhaustive bibliography 
is included, from which the reader may obtain references to more 
detailed information on every subject covered. 

5k sjt sk sk jJt 

nn 

Man cannot control travel into space until the difficult problems 
of atmospheric reentry have been solved. For many years to come, 
every manned vehicle leaving the earth's atmosphere will have to re¬ 
turn to earth, either from a decaying satellite orbit or under the more 
direct influence of the earth's gravitational field. In either case, at 
some stage of its passage through the atmosphere, the vehicle will 
travel at very high velocities. This, in turn, will result in high de¬ 
celeration and the release of large quantities of thermal energy. If 


1 


either man or sensitive equipment is to survive inside such a vehicle, 
both conditions will have to be controlled. Eventually, when more 
efficient propulsion methods have been developed, it may be possible 
to counter a space vehicle's kinetic energy by applying reverse thrust. 
That day is not yet in sight, however, and reentry remains a critical 
problem in space technology. 

The difficulty of controlling reentry is well exemplified by the 
fact that almost all meteorites entering the earth's atmosphere are 
burned up before they reach the surface of the earth. The few that do 
reach the earth retain only a small proportion of their initial mass, and 
the temperature developed inside would probably have eliminated any 
life present at the time of reentry. Meteorites have random physical 
characteristics. Most of them enter the atmosphere at speeds well 
above escape or circular velocities, and in entering the atmosphere 
they follow random trajectories. This paper is intended to show how 
man-made vehicles can be designed and controlled in flight so as to 
survive atmospheric reentry and experience no disastrous effects. 

5J: sjc >j« ^ sjc 

The mathematical analysis of reentry dynamics and thermody¬ 
namics is tremendously complex. The equations of state needed to 
obtain useful results depend upon a great many constants and variables. 
In fact, it has been impossible thus far to arrive at exact general 
relationships that could be applied to specific cases. Instead we have 
had to simplify the problem by assuming such things as an invariable 
gravitational field, disregarding the effects of the rotation of the earth, 
and so on. (These simplifying assumptions will be explained as they 
arise in this paper. ) It is believed that, despite the use of these 
assumptions, all the results are valid; many of them have already been 
experimentally verified. 

The difficulties as sociated with reentry have long been known, 
especially by astronomers, who frequently saw streaks of light in the 
night sky as meteorites burned up. The problem of reentry did not 
present itself in a practical manner until man reached a state of tech¬ 
nical development at which he could move a sizable object out of the 
atmosphere and then require it to return to the earth's surface--or to 
the surface of some other planet surrounded by an atmosphere. This 
point in the technology was reached shortly before World War II, when 
German scientists designing ballistic missiles recognized that, in 
order to achieve any appreciable range (over 250 nautical miles), the 
warhead would have to leave the atmosphere and then reenter at an 
unavoidably high velocity. 


2 


Since those days the reentry problem has been farther compli¬ 
cated by technical advances that now place mankind on the threshold of 
space travel, the successful development of long-range or interconti¬ 
nental ballistic missiles (ICBMs) and other similar projects. Thus 
reentry is no longer a subject for theoretical investigation alone; in 
this active area, theory and empirical results are blending in attempts 
to find answers to some pressing practical questions. Various branches 
of science--celestial mechanics, thermodynamics, solid-state physics 
and gas dynamics--are involved in the search for solutions to the prob¬ 
lems of atmospheric reentry. 


3 


2. FUNDAMENTAL PRINCIPLES 


2. I Celestial Mechanics . 

For all practical purposes there are essentially three types of 
uncontrolled atmospheric penetration: 

(1) Direct entry from outer space, which is often a rela¬ 
tively steep path of descent and at times almost a-straight line 

(2) The parabolic approach of a ballistic missile 

(3) The gradually descending path characteristic of a 
decaying satellite orbit. 

A number of factors affect the trajectory of a body moving above 
the surface of the earth, e.g., Coriolis force, lunar gravitation; but 
most of them can be disregarded in the study of reentry physics be¬ 
cause, at best, reentry analysis can furnish only approximate values 
for such variables as total heat transfer. Disregarding certain vari¬ 
ations in the trajectory is not likely to cause more of an error in the 
final results than failing to consider such unpredictable variations as 
wind velocity during reentry, moisture content and anomalies in air 
dens ity. 

In the first type of atmospheric penetration (direct entry from 
outer space) the vehicle would have to approach at a steep angle to 
ensure that it enters the earth's gravitational field and lands in a single 
pass. (A requirement for single-pass landing is assumed through this 
paper, unless otherwise indicated.) The vehicle's velocity for an 
entry from outer space would be at least as high as its escape velocity 
(37, 000 feet per second), or about 26, 000 feet per second in the case 
of a decaying satellite circular orbit (ref. 209). 

In its parabolic flight, a ballistic missile fails to leave the 
earth's gravitational field and does not complete an orbit around the 
earth. 


A satellite can reenter the atmosphere via several possible 
approaches: (1) a great-circle flight at small angles of inclination, 

(2) a great-circle flight at large angles of inclination, (3) a minor- 
circle flight at small angles of inclination and (4) a minor-circle flight 
at large angles of inclination. The first and third possibilities are 


4 



typical of circular orbits, while the second and fourth represent 
elliptical orbits. 

2. 2 Analysis of a Space Vehicle's Energy. 


A body approaching the atmosphere of a planet possesses a great 
amount of energy--kinetic energy generated by the body's motion and 
potential energy created by the gravitational force of the planet. 

Usually there is greater kinetic than potential energy. For example, a 
vehicle approaching the earth from outer space at escape velocity 
would have about 26, 000 Btu per pound of kinetic energy. All the 
kinetic and potential--or total--energy of the reentry body must be 
converted into some other form by the time the body comes to rest. 

The vehicle's kinetic energy at the time of reentry is -J ^ UT . 
The potential energy is equal to mgy, where y represents the altitude 
of reentry. (The reentry altitude in this paper will be taken as 3 x 10^ 
feet unless otherwise indicated. ) Therefore, 


/ - - 2 . ^ 5 " 


b. —-1-3x10 




< 3 - 


and since g can be taken as approximately 32.2 fps^ on the basis of the 
simplifying assumption that 






<3 


The total energy of the vehicle at the instant of reentry is 
approximately _ , o . \ 

~ ~ /mif -f- I o /yy) — /yv) ^ ~ ^ j 


2. 3 The Earth's Atmosphere. 


In the solution of reentry problems, a very important factor is 
the earth's atmosphere, particularly its characteristics at various 
altitudes. Our atmosphere is a mixture of various gases. At lower 
altitudes (up to about 30 kilometers from the earth's surface), these 
gases consist mainly of diatomic oxygen and nitrogen. At greater 
heights we find significant concentrations of other nuclear configura¬ 
tions. At 30 kilometers a significant concentration of ozone (O 3 ) 
builds up, and, still further, oxygen and nitrogen tend to dissociate into 
single atomic particles no longer in nuclear structure. At still higher 
altitudes, there is a significant concentration of ionized hydrogen. 
Figure 1 is a detailed breakdown of the earth's atmosphere up to cui 
altitude of 33 kilometers. 

The density distribution of the atmosphere also varies with 
altitude. The latest study available on this subject (ref. 1) indicates 


5 





Gas 

10 ^ X fraction 
(volume) 

Amount 
(Cm. S, P, T.) 

N 2 

780,900 

624,600 

02 

209 , 500 

167, 600 

A 

9 , 300 

7, 440 

COz 

300 

220 

Ne 

18 

14 

He 

5. 2 

4. 2 

CH 4 

1. 5 

1 . 2 

Kr 

1 

0 . 8 

N20 

0. 5 

0. 4 

Hz 

0. 5 

0. 4 

O 3 

0. 4 

0. 3 

Xe 

0 . 08 

0.06 

H20 

103-104 

i03-lo4 


Figure 1. Composition of dry atmosphere 

(Taken from reference 223. ) 


6 





ATMOSPHERE’S DENSITY-DISTRIBUTION CURVE 
BASED ON SATELLITE EXPERIMENTS 



7 


LOG 10 ( density in kg/M^) 




that the density is considerably greater at higher altitudes than was 
previously believed (that is, before 1956L The most recent density- 
distribution curve based on satellite experiments is shown in Figure 2. 

The temperature of the atmosphere is also an important factor 
in reentry problems. Although it varies considerably with altitude, 
the temperature has an average value of 240^K, and this is frequently 
used in reentry calculations. Employing the average temperature, it 
is possible to obtain the "atmospheric parameter," , or the recip¬ 
rocal characteristic height of the atmosphere. For earth, this factor 
is taken as j'B) * 23, 500 feet. Then, by integrating the hydrostatic 
equation, it is possible to obtain the distribution of density with altitude, 

P 

P, 

where ^ is a reference density, sometimes assumed equal to 
(ref. 1, 215). 


2. 4 Dynamics of Reentry. 

2. 4. 1 General Equations of Motion: The determination of de¬ 
celeration heating rates and other critical values associated with re¬ 
entry are based to a large degree on the fundamental equations of 
motion. It is therefore particularly important to obtain a clear under¬ 
standing of reentry dynamics. The purpose of this section is to develop 
the general equations of motion; their application to specific reentry 
problems will be treated in section 2. 4. 2. 

The general differential equations describing the motion of a re¬ 
entry body cannot be solved in the usual manner. It is necessary to 
make certain simplifying assumptions, such as disregarding the effect 
of certain variables or assuming that they remain constant during con¬ 
siderable periods of time. These assumptions can be made when 
considering specific reentry problems without serious inaccuracies. 


There are several approaches for developing the equations of 
motion, all of them based on Newtonian dynamics. For the purposes 
of this paper, two methods of developing the general equations of 
motion will be demonstrated. Using the principle of conservation of 
momentum, we have / 

oi\r ^ 


cLt 


— Sim 


0 + 


2 nnri 


(a a) 


Q- do _ 

COS& ~ ^ 




2.^ COS S> 


(C.A) 


8 










The first equation represents the momentum balance in the 
direction of motion, and the second equation, the momentum of balance 
normal to the direction of motion (ref. 215). 


A vectorial approach to the fundamental equations of motion yields 
the following: ^ u;^\. 


a = 


<i± 




L +- 


ot t 


-h 




L 


e- 


If 0 is the local flight path angle, we have 0 = —— and the 




aerodynamic force is 

^ l__cos O — i) sifvt —('j^cosO + L (9j L& 


and in Newtonian mechanics, where force equals mass multiplied by 
acceleration, we have 



- jl) CO.S & — L Si/yi (S> 


-071 


c[iJr 

d. t 




dj; 

cL t 



O', trj 



L 


5 t 



( 1 ) 

( 2 ) 


The first set of equations above represents the fundamental equa¬ 
tions of motion in terms of the absolute instantaneous velocity (tT), 
and the second set are in terms of the vertical (L/^) and horizontal 
{[T] ) velocity components. 

2. 4. 2 Solution of the Equations of Motion: To solve the equations 
of motion developed in section 2. 4. 1, certain simplifying assumptions 
must be made. They should, in general, have no serious effect on the 
calculated deceleration and convective aerodynamic heating rates, 
where they reach dominant values. Six general assumptions of this 


9 




















nature can be made, while others depend on specific reentry conditions 
such as the body's shape, the trajectory, the magnitude of the lift 
coefficient, etc. The six general assumptions are as follows: 


( 1 ) 

( 2 ) 


Both the atmosphere and earth are perfect spheres. 




cL M difl ^ 



(5) The peripheral velocity of the earth or its atmosphere 
is negligibly small compared to the velocity of reentry. 


(6) There are two possibilities of reentry: (a) without a 
lift force (or L = 0) and (b) with a lift force, but usually at a very 
small angle of reentry so that the horizontal component of lift is small 
compared to the drag. This is equivalent to saying that 

I 't(X^ O-j « I 

We now take equations of motion (1) and (2) developed in the 
section 2, 4. 1: 





Cos B — 


_L 

/yn 


S I 




From the fourth assumption we have 



dif, 



10 
















But 


chi 

n. 




<^fL 


dLuf 


'te' 




i/Tl/I 


oi u; 

Zt" 


i^iTj. 


I (3) 


• iT, t/i 

In the second equation of motion, « • n can be neglected, and we 
have 


But we have seen that as a result of the sixth assumption either 

-t- ta,^ d 


( 4 ) 


t ta^ 9-0 


or 


/ 


In any event it can be disregarded, and we have 


dij: 1 

r 

“ 

o: 

dt 1 

//m ' 


Cas 0 

1 

. (CtA 

)J 

L -1 


( 5 ) 


which is based on the relationship for drag: 


*x ^ 


where 


^fr, 


u; 


At this point we make use of a new, dimensionless variable: 

- a: -07 

iT = 


Alao from the law of gravitation: 

d. 


or 


O’ = 






S ^ 

and from the fourth assumption we can disregard the derivatives of g 
with respect to those of i/J' , and since 


dJ 7 j - dC 

di dJ 



we have 


( 6 ) 


11 































Applying this to the first equation of motion, together with the 
drag coefficient, ^ ; 


cLJ2 


S 


—j —^ = - //- 4 —~ .-S »/v\ (V-C OS 

diP 2 frn & \ 


(7) 

e) 


In order to reduce these two equations of motions into a single 
equation, a new dimensionless, dependent variable, Z, is introduced as 
follows: 

z = - •" 


on 


[T 




Ch A 


4^: ■.'< ' «?• 

‘ -n, 


( 8 ) 




where l/" is the independent variable, and ^ =: 



cl LT 


Using the well-known local-density/altitude-variation equation, 

I cLPoi 


V 




= 


we can easily develop the following relationship, which is dependent on 
the fourth assumption: 


z 


• # 


Then 


LT 

ir 

z' 


ir 

(7" 

itr 


i t 

- v 


2 


/m 


cL 
d 

Cx,A ' 

z 4 l 


lT 


— (rZ 

?^ Cose 

Substituting in equation (10) and noting that 
and — i/' \l ^ r[_ 0 



Zt 


Cl 


we have 


and 


7 ' Z i f cosG 

O' Ir ^ 


d 


lt 

-S ' 'VI (9 


lit’ 


( 10 ) 


,«• ■ 


( 11 ) 


( 12 ) 


(13) 


.12 





































Differentiating with respect to (J^ and S\^ (9 in equation (12) the 
results are 


diT, 


S d-t 


n. 


d 


LT S •'VI 


(9 


c o.:, 


( 14 ) 


and 

j_ 

dt 




cL^/ IT Z“ [rfWZ JiO-] 

dd / 


cL t 


coi (9 


Q 


ci 9 


The term ~ represents the flight-path curvature. Also from 

01 ir 

equation (12) we have: 

d <9 




d \r 


( 16 ) 


Substituting (16) and (11) into (15), the result is 


r _ LT z [- 


(17, 


From the definition of the Z function and equation (12), we note that 
equation (7) can be presented as 


(18) 


J_ d id ^ _ _ 

dt ^ dt 


I _ I . 


J - iT + 


z 


c (9 




L 


lT 




cod.e) 


Comparing equation (18) with equation (17) and noting that 




' = irz -z + ^ 


IT 


or that 


z --0 




--A 

d^ 


^ = IT Z —id -f^ I ^ 7? 


z 

lT 


13 







































(19) 


we can develop the relationship 

' Z 

ir 


Z''- z'* ~^-L 


I - 2 . 

I - LT , 4- 


U~ Z 






L 

X) 


c ox> 


9 




I- LT 

vZ 


CO, 


(9+ 


ri 


L 

Jb 


c© = O- 


( 20 ) 


It is therefore obvious that the two fundamental equations of 
motion have been reduced to a single second-order differential equation 
involving the Z function as a dependent variable and the iT function as 
the independent variable. Further, it should be noted that equation 

- LT^ 


(20) is nonlinear owing to the term 




a 


^ z 

which represents the effects of gravitation and centrifugal forces. 


Computer methods can be used to calculate the Z function in 
equation (20); also, numerical methods can be used to compute the Z 
function stepwise from this nonlinear equation. Direct integration 
methods, however, have not yet been developed. One method, probably 
not very accurate, is simple because it involves the repetition of many 
identical operations. Suppose we have an initial value of 
and the corresponding value of Z = Zj, and Z' = Z’n* Setting C ^""1 
for simplicity, we have for the second derivative 



and for the third derivative 



Using the Taylor expansion for Z^+l and Z' at the next 


point Tr » we have 

— 2. 

"7 ^ IS. 


/// 


A LT^ 


-h 


z 




( 22 ) 



•f- ( 




4 1^-+- Z^ 


/// 4 
—I-r“ 


and for Air* a fairly small value has to be selected e. g. , between 0. 001 
and 0.002. 


14 
























A large number of Z functions have been calculated for atmos¬ 
pheric reentry on a computer (ref. 48). 

Asa result of the methods used for its derivation, it is clear that 
the single equation of motion (20) will be applicable to ballistic (non¬ 
lifting) vehicles irrespective of the flight path angle at reentry, while 
in the case of lifting vehicles 

^ oj I • 

This last inequality implies that, for lifting vehicles, either the lift 
coefficient must be very small indeed or the reentry angle must be 
quite small. 


2. 4. 3 Effect of Reentry Trajectory: As we have already seen, 
there are many factors that influence the rate of heat absorption and 
deceleration of a vehicle entering the earth's atmosphere. One factor 
is the trajectory of entry, which will affect the vehicle designed to 
provide some aerodynamic lift as well as the nonlifting variety (a 
ballistic capsule). 


First let us consider the case of a nonlifting vehicle. For 
practical purposes it may be assumed that all vehicles enter the atmos¬ 
phere at about the same (escape) velocity, 26, 000 fps. Two factors 
with respect to the entering vehicle that will strongly influence heat- 
absorption rates and peak deceleration values are the entry angle 0^ 
and the value of CdA for the .particular vehicle under consideration. 

The entry angle is the minimum angle between the line of flight and any 
line in the tangential plane to the atmosphere at the point of entry. The 
point-of-entry altitude is usually taken to be about 300, 000 to 400, 000 
feet. 


Disregarding the value of 


CdA 


for the moment, it has been 


found (ref. 193) that peak deceleration is proportional to the entry angle. 
The lowest peak deceleration occurs in the case of a "grazing" entry 
resulting from a decaying satellite orbit, that is, (9^- 0. In this 
instance the peak deceleration amounts to about 9 g. The altitude at 
which this deceleration occurs, however, is controlled by the value of 

CdA 

; the lower this value, the higher the altitude of maximum 
deceleration (as will be shown later). With increasing the value 

of peak deceleration increases, at first slowly, but later more abruptly, 
as shown in Figure 3. It may be assumed that for some time space 
vehicles will not be large enough to provide any type of effective de¬ 
celeration counteraction for its occupants. Consequently the maximum 


15 







FIGURE 3 



16 


ENTRY ANGLE $p (DEGREES) AT 400,000 ft FOR Vp = 26,000 fps 









deceleration value will probably not be allowed to exceed about 10 g, 
since this is probably the maximum deceleration rate a trained pilot 
can withstand without ill effects physiologically. (This value was sug¬ 
gested by Dr. Halvey, ref. 219. ) 

As indicated on Figure 3, this value is reached at about = 
for a vehicle entering at about 26, 000 fps. 

As indicated before, the altitude at which peak deceleration 
occurs depends on the value of CpA . For a higher value of this vari- 

W 

able, the maximum deceleration will occur at a higher altitude. The 
actual magnitude of the peak deceleration, however, is not significantly 
affected--merely the altitude at which it occurs. Figure 4 shows the 
variation of vehicle velocity with altitude for three widely diverging 
values of CpA . The curves were developed by K. A. Ehricke and 

W 

R. Hermann (ref. 208, 219). Figure 4 clearly shows that, for a given 
altitude, the velocity of a vehicle with a high CpA value is much 

W 

higher than that of one with a low value, all other variables being equal 
--such as reentry velocity (assumed to be circular velocity, about 
26, 000 fps) and entry angle. This confirms tUe conclusion previously 
stated that peak deceleration for vehicles having low CpA occurs at 

W 

much lower altitudes or that the deceleration rate for such vehicles in the 
the upper regions of the atmosphere is very low. 

Another useful set of curves is shown in Figure 5. Here is a 
curve for each of a set of reentering vehicles with varying values of 
CpA , where the altitude is plotted vs. the dimensionless A ratio. 

W d' 

Ct)A 

Again we see that, for low values, peak deceleration occurs at 

lower altitudes. 

Next we have to consider reentry vehicles that have some lifting 
capacity, in other words, O . Vehicles in this class are 

limited by three considerations in particular: The first is the boundary 
imposed by maximum deceleration; again we are limited by a maximum 
of 10 g. A set of curves for various values of Cl is shown in Figure 

Cj> 

6, in which peak g is plotted vs. the entry angle. Second, the vehicle 
is limited by the maximum skin temperature of the airfoil, or lifting 
surface. Usually taken as 2500OR, this temperature is measured one 
foot from the leading edge of the airfoil, based on the equilibrium be- 
tween convective heat transfer to the airfoil and the radiative heat 


17 









FIGURE 4 

VARIATION OF VEHICLE VELOCITY WITH ALTITUDE 



1 

1 

1 

1 

1 

1 

o 

o 

o 

o 

o 

o 

o 

o 

o 

o 

o 

o 

>o 

JO 


CO 

CN 



(ij jo spuDsnoH4) - ganiinv 


18 


VELOCITY - (thousands of fps) 










400 

300 

200 

100 

0 


FIGURE 5 

ALTITUDE vs. DIMENSIONLESS 

RATIO ^ 



CpA 

W 

100.0 


10.0 



0.1 


0.01 


0.001 


0.2 0.4 0.6 0.8 1.0 

a 

T 


19 






FIGURE 7 

ALTITUDE vs. VELOCITY 


IT) 

CO 



O 

CO 



O 

CM 




UO 


O 


20 


VELOCITY - (thousands of fps) 







FIGURE 6 

PEAK g vs. ENTRY ANGLE 



6 >IV3d 


21 


ENTRY ANGLE - (DEGREES) 










transfer away from the airfoil. This state is referred to as radiation 
equilibrium temperature. The third boundary is imposed by the mini¬ 
mum aerodynamic lift required to overcome the apparent weight of the 
vehicle, that is, the difference between the real weight caused by 
gravitational attraction and the action of the centrifugal forces. Tfic 
mathematical relationship is obviously ^ 2. 

W- =■ S C lI' 






Figure 7 shows the boundaries imposed by the last two restric¬ 
tions, that is, maximum permissible skin temperature and minimum 
velocity for balancing the apparent weight of a vehicle having a value 
ly = 80 psf. From Figure 6 we can conclude that the entry angle 

for acceptable peak deceleration ( <10 g) can vary between 6^ and 9® 
for a lifting vehicle (with a maximum from 0. 25 to 3.0). This 

figure compares very favorably with the maximum entry angle of 3° 
for a nonlifting vehicle. 


22 




3. AERODYNAMIC HEATING 


The heat that is produced around a space vehicle and is then 
transferred to the vehicle's surface is one of the two most serious 
problems encountered during reentry. Several techniques of coping 
with the large quantity of heat produced have been developed (see 
section 5). However, in order to determine the most suitable method 
for cooling the surface of a particular reentry vehicle and to ensure 
that the amount of coolant is neither insufficient for the task nor so 
excessive as to result in a waste of payload capacity, it is necessary 
to predict theoretically the total amount of heat likely to be absorbed 
and the peak heating rate that may be encountered. Consequently it is 
of great importance to develop mathematical solutions for these prob¬ 
lems. 


The purpose of this section is to present useful analytical meth¬ 
ods of providing approximate values for aerodynamic heating created 
during reentry, without regard for vehicle configuration or trajectory 
during descent. The objective, therefore, is to develbp very general 
relationships between variables that affect aerodynamic heating so that 
they can be applied to specific cases. Also, it is possible to obtain 
mathematical relationships that are only suitable for limited designs 
and trajectories, as will be indicated later in one or two instances. 
Although more precise, mathematical relationships are usually of less 
value in efforts to select--and even improve on--designs for specific 
missions. The following mathematics is largely based on the general 
equations of motion and their solution (see section 2). 

The stagnation heating rate can be represented as follows 
(ref. 224): 

C lR±V" 

1/r 1/ 

In equation (23) the constants C, n and m depend on the type of 
boundary layer flow. These constants, calculated in various papers 
(e. g. , ref 241), have been found to be: 

16, 800 < C-C19, 800 
3. 1 m < 3. 22 

We can take for laminar flow n = ^ . For simplicity, m can be 
taken as equal to 3 and C as equal to 17, 000 Btu ft‘ju sec"' . The value 



/ \r 

CosQ I (23) 


23 






for C is based on the mean results obtained thus far for air at near- 
peak heating velocities, that is, ^ g 

To develop useful equations for heat, it is necessary to relate the 
factor / \ in equation (23) to the Z function. This can be done as 

\ L. ) 

follows: It was seen from equation (8) that 


z 


Therefore, 


2 






GA / 


(3 



(24) 


(3 




Z 


fL 


Cci A j j- 


(25) 


Replacing this value in equation (23), we have 

1.-#/i ffsff//A 





^6- ZC R ^ ri 


J_ 

4 


If ^ = 2_ j ajid inserting the various values for C,Psl-, etc., equa- 


( 26 ) 


/m 




tion (26) is reduced to ^ 


=590^ 





a/\R 


iT 


£ 

z 




(27) 


Cos 

The foregoing equation is based on the assumption of laminar 
flow. In the event of turbulence, the powers of [p and Z must be 
changed to 2. 2 and 0. 8, respectively. 

An analysis of equation (27) indicated that the factor 




/rn 


a /\ i9 


represents the influence on the heat flux by the configuration of the 

—X 


vehicle while the factor (iq^Q is the effect of the trajectory as de¬ 
termined by the lift/drag ratio. 

Equation (27), of course, is particularly useful in studying the 
effect of heat for vehicles which are cooled by reflectance (see section 


24 


























5). For vehicles that use cooling system of the heat-sink variety, it is 
much more important to know the total heat absorbed during reentry. 
This can be determined as follows: 



(28) 


Now going back to section 2. 4. 1 we have from equation (11) 



tr Z 

C-OS ^ 


(29) 



(30) 


and integrating 

t=- 


J 


l/T 


cos & J - f == —! 

.zrrr;— — — d. CT or L 1 - 


f 




j 


CC)S & 




IT 


Ui 

Combining the relationships in 
have the following: 




Zs/jn 


oL (T 


Q = / b, Soo 


r 


/yri 


CdA R 


J 


^2. 

quations (27), (28) and (31), we 
r ^ 

ILoLS 


^ (32) 


COS'"(9 


One particularly important result of the preceding analysis is 
to determine the effect of a changing vehicle shape on the convective 
aerodynamic heating rate. From the Reynolds number, which has 
been calculated for many reentry vehicles, it has been established that 
air flow at near-peak heating ( 0« 3 ) laminar, 

continuum-gas regime. Further, it was shown in equation (27) that for 
laminar flow the heating rate (c^^) is proportional to 



Both Cj) and Z tend to vary as the shape of the reentry vehicle changes. 
The dimensionless variable [ ^ ^ - \ is obviously an important 

factor tending to influence the heating rate. The maximum values 


25 


























of this factor are shown in the following table as they vary with changing 
lift/drag ratio: 



-0. 5 

0. 375 

-0. 25 

0. 302 

-0. 1 

0. 253 

0 

0. 218 

0. 1 

0. 184 

0. 25 

0. 138 

0. 5 

0. 098 

1.0 

0. 070 



The values fori U” /were obtained by solving for Z from 

equation (19) in section 2. 4. 2, disregarding vertical acceleration, 
vertical component of drag and by assuming that — I . These 

assumptions result in a simplified solution 



(33) 


This equation for Z is a reasonable approximation for a glide-reentry 
condition. The value of Xr for these calculations was taken as 0. 8 
when maximum heating occurs. The results shown in equation (33) are 
consistent with expectations, that is, the peak heating rate is inversely 
proportional to the lift force. In fact, if the drag force were a constant, 
the maximum heating rate would continue to decrease indefinitely with 
increasing lift. However, as the lift increases, so does the drag 
coefficient. Since 



it is found that the heating rate drops continuously up to values of 

* Beyond that there is a gradual increase in the 
heating rate for any particular shape. 

The generalized heating theory is also usefully applied in deter¬ 
mining how the drag coefficient value of a ballistic vehicle affects the 


26 

















peak heating rate. First it is necessary to obtain an approxinoation 
for Z from equation (19). By disregarding the terms of gravity, cen¬ 
trifugal force and lift and assuming that the flight-path angle is constant, 
we have _ 


Z = 1^ 





(35) 


where {Ti is some initial value of IT. Then from equation (27) for the 
ballistic case, we have 

(36) 


^.= 590 




■ LT 





z 


Since 0 is assumed to be constant for the ballistic design, it is clear 


that 






/VTL 




The conclusion is that an increase 


in , which is usually also accompanied by an increase in the radius 
of curvature of vehicle surface, causes a decrease in the maximum 
heating rate. 


Although they will not be justified mathematically, two other 
important areas are associated with reentry heating: (1) the surface 
temperature of the vehicle under radiation equilibrium and (2) the effect 
of lift/drag ratio on the total heat absorbed. 

It has been shown in several papers (ref. 216, 217, 221) that the 
maximum radiation-equilibrium temperature at laminar stagnation 
point tends to increase with increasing values of —. At the same 

CpA 

time, this temperature decreases as the value of L/D increases. 


Thus, for high-lift vehicles, the temperature tends to remain lower 
than for ballistic designs of equal weight. It may be worthwhile to 
mention a number of values associated with radiation-equilibrium 
temperature that have been obtained from computers (ref. 219) rising the 
relationship 



3 , 840 




(37) 


27 


















For these calculations, L»/D - 0. 


w 




0. 3 

2200 

4 

3100 

10 

3500 

40 

4100 


It has just been shown that the peak heating rate can be frequently 
reduced by increasing the lift/drag ratio. Increased lift prevents the 
vehicle from descending as rapidly as it would otherwise, which has 
the adverse effect of increasing the time of exposure to heating. As 
may be expected from equation (28), 



olt ci5 


a reduction in heating rate (^^)> accompanied by a substantial increase 
in the upper limit of the first integral, could easily cause an increase 
in the total amount of heat taken in by the reentry vehicle. That is, in 
fact, what happens, as may be seen from the equation developed by 
Allen and Eggers (ref. 21): 



(38) 


Here a net increase in L./D gives rise to an increase in the effective 
laminar skin-friction coefficient (Cp), because the heat is being taken 
at a higher altitude where the Reynolds number (f^) is lower and (Cp) 
is proportionately increased. 

Taking equation (32), we see that 




28 












If COi) 0 I and taking arbitrarily the upper limit iT- 0. 99, 
it is possible to calculate the values of this integral for various lift/ 
drag ratios. On the basis of equation (33), Z is taken as being equal to 

1 - Ip ^ 


CT 




L 




A table of some values thus calculated follows: 



-1. 

,0 

-0. 

5 

-0. 

25 

-0. 

1 

0 


0. 

1 

0. 

25 

0. 

5 

1. 

0 


0. 75 
0.93 
1.09 

1. 23 
1. 36 

1. 54 

1.90 

2. 53 

3. 54 


The foregoing discussion presented a practical theory of reentry 
thermodynamics and a few examples of its application. The number of 
possible applications is very large (e.g., where C03 
for glide reentry, d & / ^ 

IZt ^ ° 


for a ballistic vehicle, etc.). Some of these specialized cases have been 
solved, while others require extremely complex analysis and are best 
handled empirically or approximated by appropriate simplifying assump¬ 
tions. Some interesting discussions of specialized problems and their 
solutions may be found in references 21, 24, 26, 28, 40, 45, 54, 58, 81, 
96, 109, 114, 125, 141, 142, 200, 205, 207, 221 and 231. 


29 









4. DECELERATION 


The second most serious problem facing a reentry vehicle is that 
of deceleration. As the vehicle strikes the atmosphere in its trajectory 
toward the surface of the earth or other planet, it is possible that very 
strong forces may be developed owing to aerodynamic drag. Unless con¬ 
trolled, the drag could be considerably greater than the opposing force 
caused by the earth's gravitational field, and this could cause an ap¬ 
preciable negative acceleration. The purpose of this section is to dis¬ 
cuss the various methods of reentry, their effect on deceleration and 
methods of predicting the deceleration at any point in the trajectory of 
reentering vehicles. Again, as in the case of aerodynamic heating 
(section 3), the following mathematics is largely based on the equations 
of motion presented in section 2. 4, 1. 


The basic equation for deceleration with respect to a vehicle 
entering the atmosphere directly from a parabolic orbit can be obtained 
from the general equation of motion (section 2.4. 1): 


d tr 



By making certain assumptions, that-- 


(a) 

(b) 



Cd is constant. 


(39) 


(c) the gravitational force is small compared to drag 
and 


(d) there is no lifting force (i. e. , ballistic design), 
equation (39) reduces to 


itr _ _ tr' 

2 /m. 


(40) 


at 


JL - P 

In section 2. 3 we saw that p xL or that/ 
Then if = 








we have for the reentering vehicle 

I 


(41) 


30 









The instantaneous condition for the reentering vehicle can be 
represented as follows: y-f: ^ ( 42 ) 

where t is a very short time period (or^t) and y, is the initial altitude 
when t = 0, Therefore, for the instantaneous condition, (Tis seen to be a 
function of time, or 

_ iT t 9} ^ /Zirts:-^ 9- 

0 = e ~ Q, L 

Differentiating with respect to t, we have 

CL t 

on the basis that t/"is constant during the short time interval. Since 


(43) 


(44) 




^ n from equation (43) we have 

»r cr 0. 

cL t 


Eliminating dt between equations (40) and (45), 

cJi lT ^ 


cL O' 2 /)oo i3 cT* (T^s © 

From equation (41), — CT/^l, ’ reduces to 

cL (Ti) A cr/?z_ A PsL. 


or 


oL^ 2 y<2) cr s 

du^ A Psu 


(j- Zrrr\ 

Integrating between the appropriate limits 


Z.rrr\ fS S 

d 0" . 


r 


^ cTi A (A 


J 


kT 


J. 


cr 


Z'Tr^ (3 =• !vi © 


and 


UliC 


J 


0 LL— - 


or 


LT 




Cs^A Rsu (T 

Z nr) /3 5 ^ (9 


Then the deceleration 


zf = 


^ iCs.AP..(r 


and 


o[ tT _ 

d t 


=: — |3 l/fsc 


2r/) fZ s 1 ^ 

-2. ^ / lT 


e 
2. 


/A 

nyn (3 ^:^e- 


I 'vy 


9 


yTe 


i 






cr 


(45) 


(46) 


(47) 


(48) 


(49) 


(50) 


(51) 


31 


























Taking the escape velocity ^ differentiating the right 

side of equation (51) with respect to and equating to zero, it is 
possible to find the point of maximum deceleration, or / Jj cT 


\ clt 


MA X 


A£] ~ 

This calculation yields the result 1 J2__ 

or Lr^^^ = C). SC)! by taking (9=1, 


(52) 


Equation (52) indicates that maximum deceleration could occur 
when the velocity of vehicle reentering on a parabolic trajectory 
reaches a velocity of 0.607 , or approximately 22, 500 fps. On 

the other hand, from equation (49) we see that the velocity at any 
given altitude where (7^ assumes a definite value, say CJ^ , is 





(53) 


It is obvious from equation (53) that the actual velocity at that 
fixed altitude will depend on the value of the factor / 4 ]. yfe 




can then conclude that, for large values of Cd and small values of m 
and5 <00 Q peak deceleration will occur at higher altitudes. Since it 
is intuitively clear that high rates of heating occur when high velocities 
persist to relatively low altitudes where the atmosphere is dense, this 
result is in agreement with the conclusions formed from equation (36), 
in which we saw that 



It is also obvious from equation (53) that a shallow path (small 
angle of entry) results in peak deceleration at higher altitudes. At the 
same time it should be noted from equation (52) that the maximum 
deceleration rate is independent of vehicle characteristics ( Cd 3.nd m) 
but is influenced only by the angle of entry. 


Next, it is desirable to consider the fundamentals of deceleration 
that occurs when a vehicle enters the atmosphere from a decaying 
circular orbit. It has been established that the circular velocity 

(54) 


and 




(55) 


32 











Also, we know from section 2. 4. 1 that 



S ' 'VI 0 “f- 


/^cr^ 

2 on 



(56) 


Differentiating equation (54) with respect to time and inserting 
the result in (56), we have 

Cb i\ p _ s'/oi <9 
Z/VV1 ~ z rv 


or 


S »'Vk 


9 = 


/t 




Replacing for S / ^ ^ in equation (56) 


c^iT ^ dA/^A . /^tr^CbA 

Zt “■'3 - 




2 'Wi 


and 


since from (54) ^ == 

c/ iT ^ ^ CL /I 

__ = _^- - - 


or) 


2 ^ 


or 


^ CLA 

oL t" 2. AfY\ 


(57) 


(58) 


Equation (58) indicates that in this regime the vehicle is acceler¬ 
ating as potential energy is being converted into kinetic energy, while 
the drag is insufficient to convert any significant amount of this trans¬ 
fer energy into thermal energy. As the altitude decreases and density 
of the atmosphere increases, however, a point will be reached at which 
the acceleration is reduced to a zero value; after that, it becomes nega¬ 
tive. When the drag becomes appreciable, it is necessary to make 
certain simplifying assumptions in order to obtain an equation for de¬ 
celeration. According to reference 215, these can be summarized as 
follows: 

(1) ^ (9 c<c 

(2) LTS (9 constant 

( 3 ) 5 9 ^ 

( 4 ) Co 3 Q I 


c/cT" 

It 


33 


















From the third assumption we have that equation (56) can be 
reduced to /\ j. / \ 


Also, the second equation of motion in section 2.4. 1, 

ir “ ■ ‘ 


JlQ - cr^ 


Cc:>s 


becomes 


a Tt <§ 




LT 


cL& 


cLt ^ rL 

since it is assumed that =0* 

these equations can be combined to yield 

3 


2./yY\ Cos ^ 


lT^ 




If CT = 




'3 n. / c/.d' \ z.n^ j /3 

and integrating between limits ( 0 ^ 0~(_ ) and ( GT/ 


From equation (62) we have the deceleration of 


Jlit 

Jit 






/VI 


jT 


-I- 


c/I 

0" 


(59) 


(60) 


( 61 ) 


( 62 ) 


— I (63) 


Differentiating with respect to lT and setting the result equal to 


zero, we obtain the maximum value of 


d 


iT 


oii 


which is computed as 



This value is reached when 



0.ir34. 


It is interesting to note that, in the case of a circular-orbit 
decay, the maximum deceleration is independent of vehicle character¬ 
istics; this value is governed exclusively by the atmosphere and gravi¬ 
tational field. 


34 






























The next fundamental situation to be considered is a reentry 
vehicle have some lifting capacity. It is reasonable to make the follow¬ 
ing assumptions in this case: 


(1) Si-VI (9 <-< I 


( 2 ) 

(3) 

(4) 


Cos c9 
d(9 ^ 
dt 

<3^ 


I 


O 


-VL 


0 


(d <r 
~ddt' 


Using these assumptions, equations (56) and (59) can be reduced to 

_ __ Ch A G IT ^ 

^ t 2, /yr\ 


(64) 


and 


where 


(Tu A CT^ lT^ 

2. nY\ 





(65) 


Equation (65) can be solved forV/lso that 


ir 


i 


1 


J_ 

2 . 


jL ^ ClA/^j^<T~ 

^ 2 /TY] 


( 66 ) 


Comparing this with (/^ previously obtained and replacing in (65), 
we have r •> 


cLt 


L 




or 


d cT ^2)^ (Tb flu CT" _ _ 

iTt"” ru A fsL. (S' -f- 2 /VT) _b 

dcr 


(67) 


From equation (67) we see that ^ ^ does not reach a maximum 
value but increases, continuously approaching the asymptotic value of 

d^\ _ d ^ 


— 

dt 


MAX 


L 


( 68 ) 


35 
























As may have been expected, the characteristics of the vehicle 
exert a strong influence over the maximum deceleration value during 
descent. In fact, this value is inversely proportional to the lift/drag 
ratio. As noted from equation (68) this value is reduced to a single g 
as the lift/drag ratio approaches unity. Naturally in the selection of 
designs for a reentry vehicle, deceleration is only one of the factors 
to be considered. Although from this standpoint increasing the lift/ 
drag ratio may be desirable, it could adversely affect other aspects of 
reentry. In particular, increasing the lift/drag ratio results in a 
longer time of descent, with a corresponding increase in total heat 
transferred to the vehicle and loss of landing-point accuracy. On the 
positive side, an increase in the lift/drag ratio could improve control 
during descent and perhaps make it more comfortable for human pas¬ 
sengers. While balancing the critical reentry factors in an effort to 
obtain an optimum design, therefore, the vehicle's special purpose 
must be considered. 


In addition to the foregoing fundamental equations, containing de¬ 
celeration as the dependent variable, which were developed on the basis 
of Newtonian physics, it is desirable to relate deceleration to the Z 
function discussed in section 2. This approach is useful in that it 
eases the determination of deceleration for specific-cases, since the 
Z function has already been computed for a wide range of shallow re¬ 
entries of lifting and nonlifting vehicles of arbitrary mass and size 
having constant aerodynamic coefficients (ref. 48). 


It was seen in section 2. 4. 1 that 


CL 







(69) 


Consequently the magnitude of the acceleration vector can be repre¬ 
sented as 



In order to solve equation (70), therefore, it is necessary to 
obtain suitable relationships for the horizontal / \ and vertical 


o[ 




cit 


dt ) components of acceleration. From equation (6) (section 
4. 2), we know that, by making certain simplifying approximations. 




(71) 


36 



















and again from equation (11) 


(72) 




u-Z 

Co^& 


Substituting this value in (71), we have 




& , 



zz 

cs>s S- 


(73) 


Next it is desired to find a value for th.e vertical component of 
acceleration. It was seen in equation (18) that 


1 

da 


dt 

or that 


— i 
(/■ 

dt 




(tZ 

Cos^ (S 




ctZ 

^ (S 




Cos 9 


(74) 


CosS 


Now relating equations (73) and (74) with equation (70), we have 


dt 






A number of important practical results are indicated in the 
preceding analysis. Equation (75), of course, furnishes the absolute 
value of the deceleration for a reentering vehicle, but equation (73) is 
also very significant for shallow reentry, especially in the case of 
vehicles with an appreciable lift/drag. The reason is that much of the 
deceleration occurs at high altitudes where the angle (9 is small. Con¬ 
sequently, equation (73) can be simplified to 


dur 

(it 




(76) 


since C 0 



37 































Also, it is possible to simplify equation (75) by making use of 
inequality (3) and the six assumptions in section 2. 4. 2. The result is 
that . _ 

dij- '/dn. \rZ. ' ■ . . i- 


CL. l. 


Cos S) 



I -f ( <9 -^ 


(77) 


From this it is clear that the maximum deceleration value occurs 
when 0 CjI 0,and then we have 


d L 




ci i 


MA/ 




^ \r 


z 


MA y 


L 


JS 


(78) 


Thus we see that both L/D and vT' are strong influences over the 
maximum deceleration during reentry. Plotting / cdt Z" ] vs. lT* for 



various fixed values of L/D, it has been determined that maximum 
deceleration occurs when fj' CS±L CZ. ^ and that it tends to increase 
with decreasing values of L/D. Some peak deceleration values are 
given in the following table. 


[ 

L/D 

0,9 

1.0 

1. 6 

0. 5 

2. 7 

0. 25 

4. 9 

0. 1 

8.4 

0 

12. 7 

-0. 1 


38 




















5. COOLING TECHNIQUES 


5. I Introduction. 

As stated in section 3, the total amount of heat transferred to a 
reentering body as it passes through the atmosphere is so large and 
the heat-transfer rate is so high that this problem demands special 
attention. This consists of designing the vehicle so that reentry 
causes neither unacceptable structural damage nor temperatures in any 
part of the vehicle that exceed established limits. 

Several ways of dissipating absorbed heat during reentry have 
been considered, including (1) reflectance, that is, electromagnetic 
radiation away from the vehicle; (Z) transpiration (or chemical) cooling 
by the use of various materials, usually fluids carried inside the 
vehicle, that are spread over the surface to form a heat-absorbing 
boundary layer; (3) ablation, that is, a coating on the reentry vehicle's 
surface consisting of materials (such as ceramics) that remove heat by 
vaporizing under intense heat and pressure; and (4) the heat sink, a 
mass carried within the vehicle whose temperature rises as it absorbs 
heat. These and other possible, but unproved methods are discussed 
in this section. The information presented here is based on references 
2 through 19 and 29, 31, 43, 79, 88, 94, 124, 138, 139, 140, 143, 144, 
145, 150, 164, 190, 191, 192, 201, 206, 232, 235, 236, 238, 243, 244, 
246, 247, 253, 254, 256, 257, 259, 260 and 262. 

5. 2 Reflectance. 


The term "reflectance," as used for reentry cooling, may not be 
entirely descriptive of the function. Essentially the process of reflect¬ 
ance is the dissipation of energy by electromagnetic radiation away 
from the vehicle. Furthermore, most of the heat taken in by the vehicle 
is generated by a thermal convective process; relatively little heat is 
absorbed by radiation. Consequently "reflectance," implying a ratio of 
the reflected radiant flux to the incident radiant flux, does not apply to 
reentry cooling. Probably the term has been used because it is very 
descriptive of other phases of motion in outer space, where the primary 
source of incident energy is solar radiation and a reflectance cooling 
process is therefore essential over long periods of time. In any event, 
"reflectance," as used here, applies to all forms of cooling by radiation. 

At first glance it is obvious that reflectance is the most desirable 
method of reentry cooling, primarily because this process imposes no 


39 




weight penalty on the reentering vehicle. The several layers of radia¬ 
tive surface would probably comprise the required outer shell for the 
vehicle. Since reflectance involves no loss of material, it would be 
possible to combine the cooling system with the vehicle's normal 
structural requirements. All other methods of cooling require the 
addition of relatively heavy components to the reentry-vehicle design-- 
ablation shields, heat-sink masses, etc. 

A practical investigation of the reflectance process indicated that 
molybdenum and tungsten are probably the most suitable materials for 
this use. Thin sheets of such materials, backed by highly polished 
layers of other metals such as aluminum, could dissipate heat by radi¬ 
ation at a relatively high rate. 

Despite all the attractions of reflectance cooling, there appear 
to be a number of weaknesses in this method that have not yet been over¬ 
come. Experiments have indicated that cooling by reflectance is too 
slow in relation to the high rate of heating developed in a large-angle- 
of-entry ballistic trajectory. Also, at this time, the bonding (usually 
rubber) between layers cannot withstand temperatures much above 
ISOO^C, while reflectance cooling does not attain maximum efficiency 
until somewhat higher temperatures are reached; during ballistic re¬ 
entry, temperatures of 2500^C are not uncommon. Nevertheless, for 
reentry vehicles with Cl O^nd for certain low-angle-of-inclination 
satellite reentries, it is probable that reflectance will prove to be the 
most satisfactory cooling method, especially after more research is 
done in this area and some of the engineering difficulties are overcome. 

5. 3 Transpiration Cooling. 

It is possible to cool a heated surface by forcing a fluid through 
one or more openings in it. For bodies in relative motion with respect 
to the surrounding air, the injection of the cooling fluid should be in the 
forward region of reentry. There will be a cooling effect, first, as 
the fluid absorbs heat on reaching the surface and, then, as a protec¬ 
tive boundary layer is formed downstream from the injection point. 

This technique, known as transpiration cooling, has been studied re¬ 
cently by several investigators (ref. 30, 194, 196). Some of the work 
has been done in connection with projects to develop cooling methods 
for interior surfaces such as the jet engine, rocket-nozzle throat, etc., 
which are subject to supersonic air flow. Many of the results are also 
applicable to the cooling of reentry-vehicle surfaces. 

In the best understood process of transpiration cooling, small 
quantities of relatively cold gas are injected through a porous surface 
in the forward section of the reentry vehicle. The injection, of course. 


40 



should be applied in the region of the most severe heating, such as the 
nose of an axisymmetric body. It is also very important to know the' 
anticipated maximum reentry velocity for any given design because of 
the effect of shock waves on the downstream boundary-layer cooling. 
The wave angle and the regions where the shock waves are formed 
(such as points of discontinuity on the surface) are likely to influence 
the effectiveness of transpiration cooling for any particular design. 
Thus it may be desirable to inject only small quantities of gas in the 
stagnation section, while extending the porous surface beyond the limit 
of the shock-wave burble, in this way ensuring some boundary-layer 
effect toward the rear of the vehicle. 


Up to now, some of these problems have been disregarded in most 
research and experimentation, and attention has been concentrated on 
the decay of the cooling effect with increasing distance downstream 
from the injection region. This was based on the zero-pressure-gra- 
dient assumption; the gases used were air and helium. The results of 
this work are shown in Figures 8 and 9. 


Theoretical work on transpiration cooling has shown (ref. 30) 
that the wall-cooling parameter 

/ ~T^ ~ Tc \ 

\ 'TZus~T^ I 

is a function of the downstream distance, injection rate and Prandtl 
number. An exact mathematical relationship between these variables 
could then be established, but it would only be applicable to a flat plate 
and would be based on the assumption that no shock waves were formed 
to disturb the boundary layer. Each practical design would have to be 
tested to establish this relationship for that particular case. Further¬ 
more, since shock wave forms will vary as the velocity changes, the 
results would only hold for specific velocity regions. 


In analyzing results obtained experimentally (ref. 196) by 
measurements taken to correspond to four distance parameters where 

X - Xi 

X i 



it was found that, when plotting the wall-cooling parameter to distance- 
parameter ratio vs. wall-injection parameter ratio vs. wall-injection 
parameter on a log-log plot, the curves were, in fact, straight lines. 
Here the weill-injection parameter 



41 








42 


WALL INJECTION PARAMETER 






WALL COOLING PARAMETER 


Tw - Tc ^ 
Taw- Tc 


FIGURE 9 

SURFACE-TEMPERATURE DISTRIBUTIONS 
FOR SEVERAL INJECTION RATES 



43 







This indicates that the wall-cooling parameter is proportional to a 
power of the distance parameters, the exponent being separable into a 
product of the wall-injection parameter and a constant that depends on 
the particular gas used for cooling. This conclusion can then be sum¬ 
marized mathematically as follows: 



5. 4 Ablation. 

One of the most widely used methods for protecting the ballistic 
reentry nose cone from heat is ablation, a process in which the 
absorbed heat is dissipated as the mass of the specially constructed 
protective outer cover is lost. Heat is first absorbed by the protective 
cover acting as a heat sink. Then, when the temperature in the outer 
layer reaches the melting point, substantial amounts of heat are 
absorbed in changing the state of the ablation material. Thus, heat of 
fusion, heat of sublimation and heat of vaporization serve as important 
cooling factors. The air stream then carries the molten and vaporized 
material away from the vehicle. Much of this material then forms a 
boundary layer along the rear sections of the vehicle and so provides 
some heat protection against the free-stream conditions. Moreover, 
as most ablation materials have relatively low thermal-conductivity 
rates, the remaining ablation layers, by implication, also serve as a 
protective barrier to heat moving by conduction toward the critical 
sections of the reentering vehicle. 


The mathematics of ablation cooling is extremely simple. The 
rate of heat absorbed and remaining with the ablation surface can be 
represented as follows: 


cjQ., 

d t 






where (S) , is the amount of heat remaining, Qj. is the total heat absorbed 
by the vehicle, M is the total mass of the vehicle, m represents the 
rate of loss of mass through melting and vaporization, that is, 

_ d M ^ 

~ jLt ’ <- / and Q. the heat of vaporization. 

Although an analytical approach to ablation is possible--and 
usually profitable-before initiating a design, there are a number of 


44 











practical engineering problems that must be carefully considered, and 
then models or prototype vehicles must be extensively tested. 

Problems that should be considered include possibilities that pro¬ 
tective layers might be delaminated prematurely, thermal stresses in 
the ablation material may be excessive, and protective material in the 
envelope may be distributed unsatisfactorily. For example, it was 
found that the orientation of the glass fiber with respect to the exposed 
surface significantly influenced the occurrence of delamination. When 
the fiber was normal to the direction of heat flow, the laminating resin 
frequently decomposed before the glass melted. On the other hand, 
when the fiber was parallel to the heat flow, this did not happen. De- 
lamination of this type can be critical for a reentering vehicle because 
large chunks of ablation material will be lost before it can perform 
its mission and also because the consequently uneven outer surface 
results in turbulence, shock waves and further loss of heat-absorption 
potential. 

Among the materials used for ablation are plastics, fiber glass 
and ceramics. Most of the research on materials was done in connec¬ 
tion with reentering ballistic-missile nose cones and the resulting 
information is classified; furthermore, it would not add significantly 
to a theoretical understanding of reentry physics. Detailed information 
on ablation materials may be found in some of the references listed in 
the bibliography to this report. 

5. 5 The Heat Sink. 


One of the methods of cooling a reentry vehicle that was first 
studied was the so-called heat sink. In this method a large volume of 
material is required to absorb the heat generated by reentry. The 
heat-sink material is best distributed around the nose cone where it 
absorbs heat in the region of maximum transfer rate and so prevents 
excessive heat from reaching critical points such as instrumentation, 
electric circuits and mechanical systems. 

A material used as a heat sink must satisfy certain requirements, 
of which the most important are the possession of high thermal conduc¬ 
tivity, high specific heat and a high melting point. Of several materials 
used in heat-sink experiments, the most effective thus far is copper. 
Graphite was also considered because of its high melting point and 
other heat-absorptive qualities, but it is very hard to use owing to the 
presence of voids and other structural defects. The method considered 
for using graphite as a heat sink was to cast the material in large 
blocks and then machine the protective coat to shape. Because of the 
structural defects, however, this process has not worked well. 


45 




The dissipative capacity of a heat sink can be represented mathe¬ 
matically as follows: 

^ (T7 ) W 

5. 6 Possible New Methods of Cooling. 

In addition to those discussed before, there are several possible 
methods of cooling the surface of reentering vehicles. Some have been 
partially investigated, while others are still in the stage of speculation. 
Four of these methods are briefly described in the following subsections. 

»• 

5. 6. 1 Thermal Insulation: The objective here is to develop 
materials that, when used in coating critical areas of the vehicle's 
surface, will nignificantly retard the transfer of heat beyond the outer¬ 
most layer, The two most important qualities these materials must 
possess are very low thermal conductivity and the ability to withstand 
extremely high temperatures without melting or fracturing. 

Thermal insulation for the protection of reentry vehicles has 
been investigated to a considerable extent. Of several materials tested 
for this purpose, the most suitable have consisted of various ceramics, 
but they did not satisfy the minimum standards set by the Federal 
Government's program of reentry research. The search for more 
acceptable materials is still proceeding, and a technical breakthrough 
in this area is quite possible. 

If the right material can be found, the use of thermal insulation 
appears to have a number of advantages. In particular, it would per¬ 
mit the construction of relatively thin protective coatings for reentry 
vehicles, almost certainly a significant advantage in weight over any 
other cooling technique--possibly even reflectance. At present the 
search centers around carbide, boride, nitride and silicide cermets, 
all of which are strong and have high melting points and chemical 
stability. 

The materials tested so far have failed principally because of 
thermal shock. These ceramics have very good compression strength 
but are usually weak with respect to tension. The thermal shock causes 
unequal expansion; this results in considerable shear and tension 
stresses, which frequently cause separation and fracturing. 

5. 6, 2 Internal Cooling; Theoretically it would be possible to 
cool the surface of a reentering vehicle by circulating a coolant through 
a dense network of tubes immediately below the vehicle's surface. It 
might also be possible to keep the coolant in motion between two layers 


46 





in the outer surface. The internal cooling method would then be com¬ 
parable to systems used in the rocket nozzles of liquid-fueled engines. 

A number of technical difficulties are associated with the internal 
cooling system. The coolant would have to be circulated very rapidly 
during peak heating periods. It would be necessary to dissipate the 
absorbed heat (1) by transferring it from the fluid at some point in the 
cycle or (2) disposing of the fluid, either permanently, by dumping, 
or by temporarily removing it from the system until it cools. Either 
alternative presents certain problems. It would be difficult to accom¬ 
plish the first, while the second would greatly add to the cooling sys¬ 
tem’ s weight. 

In any case, with internal cooling, the reentry vehicle would be 
loaded with the coolant, a pump or pressure mechanism to circulate 
it and the necessary heat-transfer equipment. Moreover, the internal¬ 
cooling method is limited by the low boiling temperature of liquids as 
compared to that of solids used for reentry cooling, e.g., ablation. 

Although internal cooling appears to have a number of weaknesses, 
it is still a potentially useful method and its investigation should con¬ 
tinue. In view of the similarity of this method and the heat sink, the 
two approaches might be combined for purposes of research and testing. 

5. 6. 3 Chemical Surface Reaction: In one test of this technique, 
a wood shield was used. The wood had been soaked in water and se¬ 
cured to the vehicle so that its grain pointed away from the envelope, 
causing the air flow over the cone to be orthogonal to the grain. When 
the temperature at the surface rose high enough, the water in the wood 
was forced toward the surface where it evaporated. After about 20 
seconds of very high temperature, all the water was drained out and 
the wood started to burn. 

It is clear that this use of wood for cooling is not satisfactory, 
since the chemical reaction during the burning process would tend to 
increase the total heat generated. The use of some similar material, 
however, which would not burn but perhaps would ablate when all the 
fluid had separated as the result of chemical reaction, might be 
acceptable. Materials soaked in aqueous salt solutions were also 
investigated on the assumption that heat would be absorbed during the 
decomposition of sodium chloride and the evaporation of the water of 
crystalization. As the results have not been very conclusive, additional 
research, including the use of other materials, is probably justifiable. 

5. 6. 4 Point Mass Addition: In this method, a fluid is introduced 
at or near the stagnation point of the reentering vehicle. As the fluid 


47 




vaporizes, it absorbs heat, which is then carried away by the air 
stream. In principle, the method is similar to transpiration cooling, 
although larger amounts of fluid are ejected through a smaller area of 
the vehicle’s surface. 

5. 7 Testing and Simulation. 


As indicated in section 3, it is possible to predict with reasonable 
accuracy both heating rates and total heat created by a space vehicle 
during reentry. Further, a number of techniques for dissipating this 
heat (sections 5. 1 through 5. 6) are available. Although a theoretical 
approach to the cooling problem, which is always desirable, may 
produce a fundamental breakthrough in this difficult engineering prob¬ 
lem, it should be recognized that the effectiveness of any given tech¬ 
nique cannot be predicted with any degree of certainty. The reason for 
this is that various materials, when exposed to combinations of thermal 
shock, nonuniform expansion, strong aerodynamic forces, turbulence, 
etc. , may have unexpected reactions. Only by realistic testing can a 
reasonable assurance be obtained that a particular cooling system is, 
in fact, effective. The test objective can be attained, using full-scale 
vehicles or models, by actually propelling the vehicle beyond the 
earth’s atmosphere and observing reactions on reentry or by simulating 
reentry with appropriate equipment on the ground. 

There are dis^advantages in both methods. Experiments involving 
actual reentries pose the problem of recovering the vehicle. In addi¬ 
tion, any failure of the cooling system could easily cause extensive 
damage to the vehicle, making it impossible to establish the cause of 
failure. This, then, would prevent necessary corrective action-- 
which actually might only be minor design changes in the cooling sys¬ 
tem. It is relatively difficult to control the actual reentry of a space 
vehicle, for the trajectory may not turn out as predicted; also the 
experiment cannot be stopped at any time for the purpose of acquiring 
a better understanding of the process. 

The main disadvantage of the simulation technique is that condi¬ 
tions are never exactly comparable to those of an actual reentry. There 
are likely to be variations not only in the heating rate but in the density 
of the surrounding gas, the reentry velocity and the free stream’s 
chemical composition. All these factors significantly affect the per¬ 
formance of the cooling system. Nevertheless, some simulation tests 
are necessary to provide data that can serve as a basis for deciding 
whether to embark on full-scale reentry experiments. 

The Federal Government has developed a number of facilities 
for experimenting with reentry conditions. Since some of the detailed 


48 



specifications may still be classified for security purposes, this paper 
only discusses in general terms the principles governing the facilities' 
construction. 

In the most useful simulation testing method, rocket motors are 
used. Through proper control of the rockets, it is possible to care¬ 
fully control a number of parameters, including pressure, exhaust-gas 
composition, velocity and--most important--temperature and heating 
flux. Temperatures well over bOOG^K and heating rates of over 2000 
Btu per square foot have been reached. Velocities have varied in the 
test chambers, but they were usually below Mach 4. 

The high-intensity electric arc has been studied as a possible 
method of simulating reentry conditions, but it did not prove to be 
particularly useful. 

The reentry environment cam also be simulated in the laboratory 
by using a straight shock tube in which a removable diaphragm separates 
a gas at high pressure from the air surrounding the test model. When 
the diaphragm is very quickly removed, a shock wave is produced 
which compresses the air, heats it and sets it in motion at a velocity 
comparable to that of a reentry shock wave. Even higher pressures 
can be produced by combustion methods. The heat transfer and pres¬ 
sure can be measured with appropriate sensory equipment. This type 
of test has a limited value because of the shock wave's very short 
duration. 

Supersonic air jets have also been used in testing reentry models. 
The main advantage of using this equipment is in comparing the results 
with those obtained from rocket motors, in which the stream composi¬ 
tion is different from that normally encountered in air. 


49 


6. SPACE-VEHICLE CONFIGURATIONS 


All reentry space vehicles can be classified in two broad groups 
--those with no lift and those with L/D> 0. Each category includes 
many types and subtypes. This section briefly describes some of these 
designs and mentions a few of the advantages and disadvantages as¬ 
sociated with each. 

The nonlifting vehicle could be spherical, a spheriod or what is 
more commonly called the ballistic shape (the forward or payload 
section of a rocket or missile). The Mercury space vehicle is a typical 
ballistic shape. This design has one special advantageous quality--it 
can have a very high drag coefficient. The ballistic shape also causes 
an appreciable reduction of convective heating during reentry, which 
permits a lighter weight cooling system, e. g., ablation material. 

There are limitations, however, on the permissible drag coefficient 
for any given vehicle. Increasing the drag coefficient would cause the 
vehicle to decelerate during certain stages of reentry--and, as indi¬ 
cated previously, deceleration could be a limiting factor on the design 
because of human passengers or sensitive equipment inside the vehicle. 

Construction principles and materials suitable for use in connec¬ 
tion with ballistic configurations have been carefully investigated be¬ 
cause of their importance to the successful development of missiles. 

A considerable amount of literature on this subject is available, most 
of it protected by security classifications because of the military 
significance of such information. Good summaries of this subject may 
be found in reference 3. 

In two classes of reentry vehicles, (T . The first is a 
fixed design with a constant Ci^ for any given angle of attack. The 
second is a variable-geometry configuration. Several fixed designs 
have been tested, and a considerable volume of theoretical data is 
available, some of which has been touched upon in earlier sections of 
this report. 

The variable-geometry concept, although not new, has not been 
tested to any appreciable extent. A number of variations on this 
approach have been described in technical papers. One such proposal 
calls for folded wings on the leeward side of a relatively compact ve¬ 
hicle. After the vehicle has slowed down and reached a low altitude, 
the wings are unfolded to provide the control and stability needed for 
a soft landing. Under certain conditions it may be desirable to make 


50 



a skip reentry into the earth's atmosphere, which would call for alter¬ 
nately positive and negative lift coefficients until the velocity is lowered 
by drag forces. This kind of performance is made possible by either 
inverting the entire vehicle periodically or controlling the angle of 
attack of the lifting surfaces, producing a variable-geometry configura¬ 
tion. Another advantage of a vehicle that has an appreciable airfoil 
surface is the possibility that such a surface could be adapted to a radi¬ 
ation (or reflectance) cooling system. Thus, the lifting surface could 
serve a dual purpose and could even reduce the over-all weight require¬ 
ments for a special-purpose vehicle. 

In the design of a vehicle that is intended to reenter the atmos¬ 
phere of the earth or another planet, a number of factors must be 
taken into account. It is important, for example, to know whether the 
vehicle is to have any form of guidance system. The objective of re-- 
entry is assumed to be the arrival of the vehicle at a particular point 
pn the earth's surface or to rendezvous with another vehicle at a 
relatively low altitude in the atmosphere. If it has no guidance system, 
the vehicle must depend entirely on the forces that act upon it during 
the precalculated trajectory. Thus it is perhaps best that the vehicle 
have no lift since unexpected variations in such parameters as density 
and wind velocity are likely to affect the trajectory of a lifting vehicle 
more seriously than they would a ballistic type. On the other hand, a 
guidance and control system can correct for any unexpected deviation 
of the vehicle from the prescribed path of descent. 

How the vehicle is to be recovered is another important con¬ 
sideration. If it is to be retrieved from a body of water, its specific 
gravity must be less than 1. A vehicle intended for a soft landing on 
earth could either use a parachute to reduce its terminal velocity or 
make some kind of controlled landing, as in conventional aircraft. 

Until a vehicle's maximum reentry velocity and trajectory are 
specified, the exact form of the lifting surface cannot be prescribed. 

As the speed mounts, it becomes increasingly difficult to provide suit¬ 
able lift because of the severity of heating conditions. In these cir¬ 
cumstances, it is often necessary to compromise control requirements 
and design a more compact vehicle with a lower lift/drag ratio. Here 
is an example in which the variable-geometry configuration would be 
particularly desirable: The vehicle could have a low-lift/drag-ratio 
shape while traversing regions of severe heating; then later, when its 
speed is reduced to within safe limits and control is needed for a soft 
landing, increased lift could be provided. 

Special attention must be given to the reliability of control sys¬ 
tems and to the heating and stresses that are likely to affect their 


51 


performance. If the vehicle is required to maneuver sharply, or if 
the drag force is suddenly increased a good deal, the situation could 
become especially critical. Extremely high-temperature gas caps 
have been known to form around control surfaces. A reasonable margin 
of safety, therefore, should be built into every control system, partic¬ 
ularly in the case of vehicles that are expected to carry humans. 


52 


7. CONCLUSIONS 


The physics of atmospheric reentry is an essential part of modern 
space science. Although it would be desirable to perfect the theoretical 
aspects of this subject, the practical requirements of space travel will 
be satisfied by approximate results and empirically verified principles. 
This acceptance of something less than perfection is fortunate, because 
the field of reentry mechanics and thermodynamics is extremely com¬ 
plex. 


Up to now it has been impossible to develop mathematically ideal 
solutions for variables--such as heating rate and peak deceleration-- 
that could be obtained for any reentry condition merely by inserting 
appropriate values in a general equation. Instead, it has been neces¬ 
sary either to sacrifice accuracy by using certain simplified general 
equations or, if greater precision is needed, to develop specialized 
equations applicable only to a very narrow range of reentry conditions. 

As further research is performed, some of these deficiencies will 
probably be overcome. In fact, one noted scientist stated recently that 
he is now completing work on . a second order theory to provide a 
single analytical approximate solution which can be applied to the entire 
region of entry." The objective, of course, is to improve on the ac¬ 
curacy of previously developed general solutions. Whether or not this 
particular effort proves successful, there is little doubt that we can 
expect major steps forward in understanding reentry mechanics, pro¬ 
vided that this effort continues to receive necessary support. 

In addition to the general conclusion just briefly stated, it is pos¬ 
sible to reach a number of specific conclusions on the basis of the in¬ 
formation presented in this paper; they can be summarized as follows: 

(1) Vehicles reentering the earth’s atmosphere from space 
are expected to travel at high velocities. The lowest probable reentry 
velocity would be that of an earth satellite in a decaying orbit, which 
would be over 25, 000 feet per second. 

(2) Much of the energy contained in a reentering vehicle by 
virtue of its altitude and velocity must be converted into heat before it 
can make a soft landing on the earth's surface. 

(3) For purposes of reentry mathematics, it may be as¬ 
sumed that the earth's atmosphere varies exponentially with altitude. 


53 


In order to solve numerical problems it may be assumed that a body 
enters the atmosphere when it drops to an altitude of about 300, 000 to 
400, 000 feet. 

(4) The trajectory of a reentry vehicle is a very important 
factor in controlling such variables as deceleration and heating. An¬ 
other important factor is the lift/drag ratio. Although increasing this 
ratio tends to reduce peak deceleration value and peak heating, it has 
the undesirable effects of increasing total heat and reducing landing 
accuracy. The trajectory and design of a vehicle can only be selected 
on the basis of the specific missions for which it is intended. Two 
important considerations must be kept in mind at all times: (1) The 
surface temperature has to be kept below a certain maximum value, 
which will depend on the cooling technique used. (2) The peak de¬ 
celeration should be limited. If there are humans in the vehicle, the 
peak deceleration should be held below 10 g; otherwise, the sensitivity 
of equipment on board determines the maximum permissible decelera¬ 
tion. 


(5) Considerable attention must be devoted to the cooling 

t 

techniques used with various reentry vehicles. Insufficient amounts of 
cooling material could well cause disaster during reentry, while an 
excessive amount would reduce the payload capacity for other purposes 
Moreover, any improvement in cooling techniques, such as the develop 
ment of greater efficiency and the reduction of weight, would have 

obvious advantages. 

* 

(6) Every cooling technique should be carefully tested 
before it is used on a space vehicle. There might be all kinds of un¬ 
expected developments, that could cause failure--the bonding material 
is often unsuitable; thermal shock could have serious consequences. 
Before a cooling technique is accepted, it would be desirable to conduct 
both ground tests and tests under actual reentry conditions. 

(7) Reflectance holds the greatest promise of efficient 
cooling, but much more work remains to be done before this technique 
would be suitable for most reentry-vehicle designs. At present, the 
outstanding weakness of this method is that reflectance alone cannot 
dissipate heat fast enough under certain reentry conditions. 

(8) In the current state of the art, ablation is the most 
satisfactory cooling technique for most reentry vehicles. The ablation 
shield absorbs heat through fusion, sublimation and vaporization. In 
addition, the residue of the ablation material forms'a protective bound¬ 
ary layer along the rear sections of the vehicle. 


54 


(9) The use of gas for transpiration cooling is not the 
most efficient process. One way to improve this method would be to 
use a liquid in such a way that vaporization occurred in the porous 
region just below the surface. There would be an additional cooling 
effect from the fluid's absorption of the heat of vaporization. It is 
important that only gas reach the surface. Otherwise turbulence would 
be caused, and that would lessen the cooling effect of gas in the 
boundary layer downstream from the injection point. The only way to 
achieve this would be by extremely sensitive injection control, based 
on surface temperature. Additional research and experimentation in 
connection with transpiration cooling techniques would probably be 
very worthwhile. 

(10) All reentry vehicles can be classified in two broad 
groups--they can have a lift/drag ratio other than zero or no lift at all. 
The selection of a design largely depends on the vehicle’s mission, as 
well as other factors. One of the most promising configurations is 
that of the variable-lift/drag-ratio vehicle. Here the value of the lift 
can be modified to gain maximum benefit of velocity, air density and 
other similar variables. It may thus be possible to improve on the 
control and guidance of the vehicle and to reduce the hazards of de¬ 
celeration and heating as well. 


55 



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64 




































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66 






























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106. Levy, L. L. , Jr. An Approximate Analytical Method for Studying 

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107. Lichtenstein, R. H. Analytical Investigation of the Dynamic Be¬ 

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109. Lieblein, S. Analysis of Temperature Distribution and Radiant 

Heat Transfer Along a Rectangular Fin of Constant Thickness. 

NASA TN D-196, November 1959. 

110. Low, G. M. Nearly Circular Transfer Trajectories for Descend¬ 

ing SateUites. NASA TR R-3, 1959. 

111. Luidens, R.W. Approximate Analysis of Atmospheric Entry 

Corridors and Angles. NASA TN D-590, January 1961. 

112. McFall, J. C. Free-Flight Drag Measurements of Rocket- 

Boosted Models of Two Reentry Body Configurations at Mach 

Numbers from 4. 3 to 0. 6. NASA TM X-118, October 1959, 

Confidential. 

113. McGehee, J. R. , and Hathaway, M. E. Landing Characteristics 

of a Reentry Capsule with a Torus-Shaped Air Bag for Load 

Alleviation. NASA TN D-628, November I960. 

114. Matting, F. W. , Chapman, D. R. , Nyholm, J. R. , and Thomas, 

A. G. Turbulent Skin Friction at High Mach Numbers and 
Reynolds Numbers in Air and Helium. NASA TR R-82, I960. 


67 






























115. 


Mayhue, R. J. Free Flight Measurements of the Base Pressures 
and Drag of a Flare-Stabilized Cylindrical Reentry Body with an 

Elliptical Blunt Nose at Mach Number s from 1. 9 to 0. 7. NASA 
TM X-309, September I960, Confidential. 

116. Mayhue, R. J. , and Blanchard, W, S. Free Flight Investigation 

of the Base Pressure and Drag of a Flare-Stabilized Blunt-Nose 

Reentry Body Having a Fineness Ratio of 3. 11 at Mach Numbers 

from 0.70 to 1.90. NASA TM X-214, March I960, Confidential. 

117. Moeckel, W. E. Departure Trajectories for Interplanetary 

Vehicles. NASA TN D-80, November 1959. 

118. . Trajectories with Constant Tangential Thrust in Central 
Gravitational Fields. NASA TR R-53, I960. 

119. Morris, O. A. , and Keith, A. L. , Jr. Experimental Investigation 

of Pressure Distribution and Static Aerodynamic Characteristics 
for Three Supersonic-Impact Ballistic Reentry Shapes at Mach 
Numbers of 1.93, 2.55, and 3.05. NASA TM X-453, March 
1961, Confidential. 

120. Mugler, J. P. , Jr., and Olstad, W.B. Static Longitudinal 

Aerodynamic Characteristics at Transonic Speeds of a Blunted 
Right Triangular Pyramidal Lifting Reentry Configuration for 
Angles of Attack up to IIQO. NASA TN D-797, June 1961. 

121. . Static Longitudinal Aerodynamic Characteristics at 
Transonic Speeds of a Lenticular-Shaped Reentry Vehicle. 

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122. Nielsen, J. N. , Goodwin, F. K. , and Mersman, W.A. ’’Three- 

Dimensional Orbits of Earth Satellites Including Effects of 
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123. Olstad, W. B. , Mugler, J. P. , Jr., and Cahn, M. S. Static 

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Pyramidal Lifting Reentry Configuration at Transonic Speeds. 

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124. O’Neal, R. L. and Rabb, L. Heat-Shield Performance During 

Atmospheric Entry of Project Mercury Research and Develop¬ 

ment Vehicle. NASA TM X-490, May 1961. 


68 
































125. 


Page, W.A., Canning, T. N. , Craig, R. A. , and Stephenson, J, D. 
Measurements of Thermal Radiation of Air from the Stagnation 

Region of Blunt Bodies Traveling at Velocities up to 31, 000 Feet 

per Second. NASA TM X-508, June 1961, Confidential. 

126 . Paulson, J. W. , Shanks, R, E. , and Johnson, J. L. , Jr. Low- 

Speed Flight Characteristics of Reentry V ehicles of the Glide- 

Landing Type. NASA TM X-331, September I960, Confidential. 

127. Pearson, A.O. Wind-Tunnel Investigation at Mach Numbers from 

0. 20 to 1.17 of the Static Aerodynamic Characteristics of a 
Possible Reentry Capsule. NASA TM X-262, March I960, Conf. 

_. Wind Tunnel Investigation at Mach Numbers from 0. 40 

to 1. 20 of the Static Aerodynamic and Control Characteristics of 
a Model of a Nonlifting Reentry Capsule in Combination with a 

Rocket Booster. NASA TM X-317, September I960, Confidential. 

129. . Wind-Tunnel Investigation at Mach Numbers from 0. 6 to 
1. 2 of the Static Aerodynamic Characteristics of a Model of a 
Possible Nonlifting Reentry Capsule in Combination with a 
Rocket Booster. NASA TM X-318, September I960, Confidential. 

130. Penland, J.A., and Armstrong, W, O. Static Longitudinal Aero¬ 

dynamic Characteristics of Several Wing and Blunt-Body Shapes 
Applicable for Use as Reentry Configurations at a Mach Number 

of 6. 8 and Angles of Attack up to 90®. NASA TM X-65, October 

1959, Confidential. 

131. Petynia, W. W. Model Investigation of Water Landing of a Winged 

Reentry Configuration Having Outboard Folding Wing Panels. 

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132. Phillips, W. H. Research on Blunt-Faced Configurations at Angles 

of Attack Between 60® and 90®. NASA TM X-315, September 

1960, Confidential. 

133. Polhamus, E. C. , and Geller, E. W. Pressure and Force 

Characteristics of Noncircular Cylinders as Affected by Reynolds 
Number with a Method Included for Determining the Potential 
Flow about Arbitrary Shapes. NASA TRR-46, 1959. 

134. Rainey, R. W. Summary of Aerodynamic Characteristics of Low- 

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69 






































135. 


Rainey, R. W, Working Charts for Rapid Prediction of Force and 
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136. Rainey, R. W. , and Close, W. H. Studies of Stability and Control 

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137. Rakich, J. V. Supersonic Aerodynamic Performance and Static- 

Stability Characteristics of Two Blunt-Nosed Modified 13^ Half- 

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138. Rashis, B. Exploratory Investigation of Transpiration Cooling of 

a 40^ Double Wedge Using Nitrogen and Helium as Coolants at 
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139. Rashis, B. , and Hopko, R. N. Analytical Investigation of Abla- 

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140. Rashis, B. , and Walton, T.E. An Experimental Investigation of 

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141. Reller, J. O. Heat Transfer to Blunt Axially Symmetric Bodies. 

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142. Reller, J. O. , and Seegmiller, H. L. Convective Heat Transfer 

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143. Roberts, L. An Analysis of Ablation-Shield Requirements for 

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144. _. An Analysis of Nose Ablation for Ballistic Vehicles. 

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145. . Stagnation Point Shielding by Melting and Vaporization. 

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146. Robinson, R. B. , and Spearman, M. L. Stability and Control. 

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70 































Ru.msey, C.B., and Lee, D. B, Heat Transfer Measurements on 
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148. Rumsey, C. B. , Piland, R. O, , and Hopko, R. N. Aerodynamic- 

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149. Sarabta, M.F. Aerodynamic Characteristics of a Blunt Half- 

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150. Savin, R. C. , Gloria, H. R. , and Dahms, R. G. Ablative Prop- 

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151. Scallion, W.I. Full-Scale Wind-Tunnel Investigation of the Low- 

Speed Static Aerodynamic Characteristics of a Model of a Re¬ 
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152* Shaw, D. S., and Turner, K. L. Wind-Tunnel Investigation of 
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153. Slye, R. E. An Analytical Method for Studying the Lateral Motion 

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1960. 

154. Smith, D. W. , and Walker, J. H. Skin Friction Measurements in 

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155. Smith, F. M. , and Nichols, F. H. A Wind-Tunnel Investigation of 

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156. Smith, O. E. , and Chenoweth, H. B. Range of Density Variability 

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157. Smith, W. G. A Wind-Tunnel Investigation at Subsonic and Low 

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158. Sommer, S. C. , Short, B. J. , and Compton, D. L. Free-Flight 

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159. Spar row y • TJn.stG^dy St5>^ri3.tion."*JPoixit S 

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160. Sp encer, B. An Investigation at Subsonic Speeds of Aerodynamic 

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161. _. A n Investigation at Subsonic Speeds of the Longitudinal 

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162. . An Investigation of the Aerodynamic Characteristics at 
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163. Stainback, P. C. Visual Technique for Determining Qualitative 

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164. Stephens, E. W. Afterbody Heating Data Obtained from an ATLAS- 

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165. Swann, R. T. , and South, J. A Theoretical Analysis of Effects of 

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166. Swenson, B. L. A Study of Methods for Simulating the Atmosphere 

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72 




























167. Swenson, B. L. Exploratory Study of the Reduction in Friction 

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% * 

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170. Trescot, C.D., and Putnam, L. E. Low-Speed Full-Scale 

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171. Tunnell, P. J. The Static and Dynamic Stability Derivatives of a 

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180. Weston, K. C. , and Swanson, J. E. A Compilation of Wind- 

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188. Wong, T.J, , and SI ye, R. E. The Effect of Lift on Entry 

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Institute of Aeronautical Sciences Paper No. 60-8, 25-27 
January I960. 

260. Yaffee, M. ’’Ablation Wins Missile Performance Gain.” 

Aviation Week. 18 July I960. 

261. Zartarian, G. , Hsu, P. T. , and Ashley, H. ’’Dynamic Airloads 
and Aeroelastic Problems at Entry Mach Numbers.” Institute of 
Aeronautical Sciences Paper No. 60-32, 25-27 January I960. 

National Aeronautics and Space Administration: 


262. Welver, J. E. Comparison of Theoretical and Experimental 

Valves for the Effective Heat of Ablation of Ammonium Chloride. 
NASA TN D-553, November I960. 


81 











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appendix I 




Cl 



C 


C, 

C 

Ci. 

C, 

E 

esc 

F 


* 

U 




Notation 

acceleration 
acceleration vector 
area of cross section 
specific heat 
heat of fusion 

dimensional constant for stagnation-point heat-transfer rate 

heat of vaporization 

drag coefficient 

lift coefficient 

drag force 

basis of a natural logarithm 

energy 

escape 

force vector 

gravitational acceleration 

unit vector in the X direction of a Cartesian coordinate system 
unit vector in the /^direction 
unit vector in the & direction 

unit vector in the y direction of a Cartesian coordinate system 
unit vector in the Z direction of a Cartesian coordinate system 


k 


83 




Ih 

R 

Re 

Ro 

$ 


ratio of heat rate (at a given point on the surface to the 
heating rate ((^^) at the stagnation point or >r<^ - 
constant in transpiration cooling equation 
lift force 

mass or rate of mass loss 

total mass in the reentry vehicle 

mean molecular weight of atmosphere 

wall-injector parameter 

Prandtl number 

dynamic pressure 

aerodynamic heating rate per unit area 

aerodynamic heating rate per unit area at stagnation point 
total heat absorbed 

amount of heat absorbed and retained in the ablation shield per 
unit mass 

total heat absorbed by the ablation shield per unit mass, or 
total heat absorbed by the heat sink 
radius (usually distance to the center of the earth) 
radius of curvature of a space-vehicle surface configuration 
Reynolds number 
radius of earth 
airfoil surface area 
sea level 


84 


time 



ir 

ir 


t/'i- 

IV 

)( 

/3 


temperature 
temperature of fusion 
reentry temperature 

adiabatic wall temperature under zero coolant injection 

temperature of the coolant 

temperature at the wall 

stagnation surface temperature 

free-stream atmospheric temperature 

velocity of reentering body 

' 

dimensionless variable equal to —~- 
circular velocity 
escape velocity 

velocity in the direction of motion or tangential component of 
circumferential velocity 
velocity vertical to tangential component 
mass of the vehicle in pounds 
distance from stagnation point 

distance from stagnation point to start of injection 
distance from stagnation point to injection cutoff point 
distance parameter 
altitude 

reciprocal characteristic height of the atmosphere 


85 





ratio of specific heats 


£ 

5 


surface radiative emissivity 
local flight path angle 


Of angle of entry into the atmosphere 


/ 


U. 


coefficient of viscosity 


SL. coefficient of viscosity at sea level 




/O 


u 




CO 




density 

a reference atmospheric density frequently taken as equal to 
density at sea level 
density at altitude y 
free-stream atmospheric density 

mainstream mass flow per unit area at the injection cutoff 


point 


(/^IT^iaT mean injection mass flow per unit area 


cr 


ratio 




A 


jjg 


Btu 


British thermal unit 


cep circular probable error 


fps 


feet per second 


ICBM intercontinental ballistic missile 


psf pounds per square foot 


86 



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